The Equation of Transfer

dIν = jν(ρ/μ) dz - Κν Iν (ρ/μ) dz Basic Ideas Change = Source - Sink Change In Intensity = Emitted Energy -“Absorbed” Energy dIνdνdμdtdA = jν(ρdAds)dνdμdt -ΚνIν(ρdAds)dνdμdt dIν = ρjνds - ρΚνIνds dIν = jν(ρ/μ) dz - Κν Iν (ρ/μ) dz μ dIν / ρdz = jν- ΚνIν Remember: dτν = -ρΚνdz μ(dIν/dτν) = Iν – Sν Where Sν ≡ jν / Κν Z dz θ ds The geometry gives: dz = μds Equation of Transfer

The Source Function Sν Κν is the total absorption coefficient Κν = κν + σν The equation of transfer is μ(dIν/dτν) = Iν - Sν The Emission Coefficient jν = jνt + jνs If there is no scattering: jνs = 0 and σν = 0 Then jν = jνt and Κν = κν Thus Sν = jν/ Κν = jνt / κν = Bν(T) In the case of no scattering the source function is the Planck Function! Equation of Transfer

The Slab Redux Slab: No Emission but absorption μdIν/ρdz = - ΚνIν dIν/Iν = dτν/μ I(0,μ) = I(τν,μ)e-τ/μ Note that the previous solution was for μ = 1 which is θ = 0. Equation of Transfer

Formal Solution μdI/dτ = I - S : Suppress the frequency dependence It is a linear first-order differential equation with constant coefficients and thus must have an integrating factor: e-τ/μ d/dτ (Ie-τ/μ) = -S e-τ/μ / μ dI/dτ e-τ/μ + I e-τ/μ (-1/μ) = -S e-τ/μ / μ dI/dτ + I/-μ = -S/μ μdI/dτ = -S + I Equation of Transfer

Equation of Transfer

Boundary Conditions For the optical depths: τ1 = 0 and τ2 → ∞ We Require: The above is the lower boundary “boundedness” condition. The upper boundary condition is that there be no incoming radiation. Equation of Transfer

Physical Meaning The physical meaning of the solution is that the emergent intensity is the weighted mean of the source function with the weights proportional to the fraction of energy getting to the surface at each wavelength / frequency. The solution depends on the form of S: Let S = a+bt Then I(0,μ) = a + bμ Equation of Transfer

Separate τ Into Domains Incoming and Outgoing at an Arbitrary Depth

Keep the Signs and Angles Straight Out Θ: 0 to 90 degrees μ: 1 to 0 Integration: τ to ∞ In Θ: 90 to 180 μ: 0 to -1 Integration: τ to 0 Z dz θ ds Equation of Transfer

What Is the Difficulty? Consider the Source Function S These appear simple enough but … Consider the Source Function S What is the “Usual” form for S? For coherent isotropic scattering (a very common case): Sν = (κν/(κν + σν)) Bν + (σν /(κν + σν)) Jν So the solution of the equation of transfer is: But J depends on I: Jν = (1/4π)Iνdμ Therefore S depends on I This is not an easy problem in general! Equation of Transfer

Schwarzschild-Milne Equations Let us look at the Mean Intensity Jν In the second integral we have 1) reversed the order of integration so μ => -μ and 2) changed the sign on both parts of the exponent. (See Slide 8). Equation of Transfer

Continuing ... For integrand 1: w = μ-1 => dw = -μ-2dμ so dw/w = -dμ/μ. For the second w = -μ-1 for which dw/w = -dμ/μ also. In the first integral we also changed the order of integration (which cancelled the minus sign on dw/w). Equation of Transfer

The Exponential Integral The Integrals in J(τ) are Called the Exponential Integrals These integrals are recursive but that is NOT the recursion above! Equation of Transfer

Schwarzschild-Milne Equations The Mean Intensity Equation is called the Schwarzschild Equation and the flux equation is called the Milne equation. The surface flux is: Equation of Transfer